In a commutative semigroup \(\left(S,+\right)\) , a set is said to be
an \(IP\)  set if it contains a configuration of the form \(\left\lbrace  \sum _{t\in H}x_{t}:H\text{ is nonempty finite subset of }\text{$\mathbb {N}$}\right\rbrace  \) 
for some sequence \(\langle x_{n}\rangle _{n\in \mathbb {N}}\) , where
it is called an \(IP_{r}\)  set if it contains configuration of the
form \(\left\lbrace  \sum _{t\in H}x_{t}:H\subseteq \left\lbrace  1,2,\ldots ,r\right\rbrace  \right\rbrace  .\) 
A set is said to be \(IP^{\star }\) and \(IP_{r}^{\star }\)  respectively
if it intersects with every \(IP\)  sets and \(IP_{r}\)  sets resp. For
details the readers may see the book [1]}.
